Let $2=p_1,p_2,\cdots ,p_n$ be the first $n$ prime numbers.
Suppose $N$ is a natural number and that $A=\{a+1,\cdots, a+N\}$ be a set of $N$ consecutive integers.
Let $P_n=p_1\cdot p_2\cdot \cdots p_n$ and $G_A=\{x:x\in A : \gcd(x,P_n)=1\}$
What is the maximum value of $|G_A|$?
In other words:At least how many integers from the set $A$ will be sieved by the first $n$ primes?
I would like to see (if it is possible) an elementary method-result .
Thanks in advance!
We don't understand this situation as well as we would like to, particularly when $A$ is between say $n$ and $n^2$. It's related to the Jacobsthal function on primorials, if you want to do some digging around.